Here are the solutions for the special right triangles worksheet. I have calculated the missing sides using the properties of 45-45-90 and 30-60-90 triangles.
1.
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Type: 30-60-90 triangle. The side opposite the 30° angle is 7.
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Rule: Hypotenuse = $2 \times$ (short leg). Long leg = (short leg) $\times \sqrt{3}$.
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Hypotenuse: $7 \times 2 = 14$
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Long Leg: $7\sqrt{3}$
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Answer: Legs: $7, 7\sqrt{3}$; Hypotenuse: $14$
2.
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Type: 45-45-90 triangle. One leg is 6.
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Rule: Both legs are equal. Hypotenuse = leg $\times \sqrt{2}$.
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Other Leg: 6
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Hypotenuse: $6\sqrt{2}$
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Answer: Legs: $6, 6$; Hypotenuse: $6\sqrt{2}$
3.
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Type: 45-45-90 triangle. Hypotenuse is $34\sqrt{2}$.
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Rule: Leg = Hypotenuse $/ \sqrt{2}$.
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Legs: $34\sqrt{2} / \sqrt{2} = 34$
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Answer: Legs: $34, 34$; Hypotenuse: $34\sqrt{2}$
4.
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Type: 30-60-90 triangle. Short leg (opposite 30°) is 5.
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Rule: Hypotenuse = $2 \times$ short leg. Long leg = short leg $\times \sqrt{3}$.
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Hypotenuse: $5 \times 2 = 10$
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Long Leg: $5\sqrt{3}$
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Answer: Sides: $5, 5\sqrt{3}, 10$
5.
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Type: 30-60-90 triangle. Long leg (opposite 60°) is $6\sqrt{3}$.
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Rule: Long leg = short leg $\times \sqrt{3}$. So, short leg = 6.
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Short Leg: 6
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Hypotenuse: $6 \times 2 = 12$
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Answer: Sides: $6, 6\sqrt{3}, 12$
6.
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Type: 30-60-90 triangle. Hypotenuse is 8.
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Rule: Short leg = Hypotenuse $/ 2$. Long leg = short leg $\times \sqrt{3}$.
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Short Leg: $8 / 2 = 4$
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Long Leg: $4\sqrt{3}$
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Answer: Sides: $4, 4\sqrt{3}, 8$
7.
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Type: 45-45-90 triangle. Hypotenuse is $10\sqrt{2}$.
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Rule: Leg = Hypotenuse $/ \sqrt{2}$.
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Legs: $10\sqrt{2} / \sqrt{2} = 10$
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Answer: Legs: $10, 10$; Hypotenuse: $10\sqrt{2}$
8.
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Type: 30-60-90 triangle. Long leg (opposite 60°) is $8\sqrt{3}$.
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Rule: Long leg = short leg $\times \sqrt{3}$. So, short leg = 8.
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Short Leg: 8
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Hypotenuse: $8 \times 2 = 16$
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Answer: Sides: $8, 8\sqrt{3}, 16$
9.
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Type: 45-45-90 triangle. Hypotenuse is 25.
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Rule: Leg = Hypotenuse $/ \sqrt{2}$. To rationalize, multiply top and bottom by $\sqrt{2}$.
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Legs: $\frac{25}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{25\sqrt{2}}{2}$ (or $12.5\sqrt{2}$)
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Answer: Legs: $\frac{25\sqrt{2}}{2}, \frac{25\sqrt{2}}{2}$; Hypotenuse: $25$
10.
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Type: 30-60-90 triangle. Short leg (opposite 30°) is 9.
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Rule: Hypotenuse = $2 \times$ short leg. Long leg = short leg $\times \sqrt{3}$.
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Hypotenuse: $9 \times 2 = 18$
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Long Leg: $9\sqrt{3}$
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Answer: Sides: $9, 9\sqrt{3}, 18$
Final Answer:
1. Legs: $7, 7\sqrt{3}$; Hypotenuse: $14$
2. Legs: $6, 6$; Hypotenuse: $6\sqrt{2}$
3. Legs: $34, 34$; Hypotenuse: $34\sqrt{2}$
4. Sides: $5, 5\sqrt{3}, 10$
5. Sides: $6, 6\sqrt{3}, 12$
6. Sides: $4, 4\sqrt{3}, 8$
7. Legs: $10, 10$; Hypotenuse: $10\sqrt{2}$
8. Sides: $8, 8\sqrt{3}, 16$
9. Legs: $\frac{25\sqrt{2}}{2}, \frac{25\sqrt{2}}{2}$; Hypotenuse: $25$
10. Sides: $9, 9\sqrt{3}, 18$
Parent Tip: Review the logic above to help your child master the concept of right triangle practice worksheet.
